Question

Eliminate $\theta $ from the relations: $5x = 3\sec \theta $ and $y = 3\tan \theta $.

Answer 1

Hint: In the given problem, we are given two equations involving the trigonometric functions. The question requires thorough knowledge of trigonometric functions, formulae and identities. We solve the two equations using the method of elimination to find the values of trigonometric functions. Then, we use some trigonometric equations and formulae to get to the result.

Complete step-by-step answer:
In the given question, we are provided with the equations: $5x = 3\sec \theta $ and $y = 3\tan \theta $.
So, we have to find the values of both the trigonometric functions: tangent and secant using the two equations.
Consider the equation, $5x = 3\sec \theta $
Shifting the terms from one side to another to find the value of secant, we get,
$ \Rightarrow \sec \theta = \dfrac{{5x}}{3}$
Now, from equation $y = 3\tan \theta $, we have the value of tangent as,
$ \Rightarrow \tan \theta = \dfrac{y}{3}$
Now, we know the trigonometric identity ${\sec ^2}\theta = 1 + {\tan ^2}\theta $.
So, we substitute the values of secant and tangent in the identity, we get,
\[ \Rightarrow {\left( {\dfrac{{5x}}{3}} \right)^2} = 1 + {\left( {\dfrac{y}{3}} \right)^2}\]
Computing the squares, we get,
\[ \Rightarrow {\dfrac{{25x}}{9}^2} = 1 + {\dfrac{y}{9}^2}\]
Hence, the $\theta $ has been eliminated from the two relations.

Note: There are six trigonometric ratios: $\sin \theta $, $\cos \theta $, $\tan \theta $, $\cos ec\theta $, $\sec \theta $and $\cot \theta $. Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$, ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above. Algebraic operations and rules like transposition rule come into significant use while solving such problems.